2024 NJC P2 Q6

For a positive integer \(n\), it is given that \(\mathrm{P}\left( A=n \right)=0.009542\) and \(\mathrm{P}\left( A=n+1 \right)=0.004090\), where \(A\sim\mathrm{B}\left( 2n+1,p \right)\). Show that \(p\) satisfies an equation of the form \(\frac{p}{1-p}=k\), where \(k\) is a constant to be determined. Hence find the value of \(n\) and the variance of \(A\).[4]